Gravitational equilibrium in the presence of a positive cosmological constant

We reconsider the virial theorem in the presence of a positive cosmological constant. Assuming steady state, we derive an inequality of the form >~A (/8G₍) for the mean density of the astrophysical object. The parameter A depends only on the shape of the object. With a minimum at Aₒ₏₇₄ₑ₄=2, its value can increase by several orders of magnitude as the shape of the object deviates from a spherically symmetric one. This indicates that flattened matter distributions such as, e. g. , clusters or superclusters, with low density, cannot be in gravitational equilibrium.